Hydrostatic Pressure on Surfaces

7/27/2019 Hydrostatic Pressure on Surfaces

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where is the depth of liquid above the centroid of the submerged area.

Derivation of Formulas The figure shown below is an inclined plane surface submerged in a liquid. The total area of theplane surface is given by , cg is the center of gravity, and cp is the center of pressure.

The differential force acting on the element is

From the figure ,

Integrate both sides and note that and are constants,

Recall from Calculus that


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From the figure, , thus,

The product is a unit pressure at the centroid at the plane area, thus, the formula can beexpressed in a more general term below.

Location of Total Hydrostatic Force (Eccentricity) From the figure above, is the intersection of the prolongation of the submerged area to the freeliquid surface. Taking moment about point .

Where

Again from Calculus, is called moment of inertia denoted by . Since our referencepoint is ,

Thus,

By transfer formula for moment of inertia , the formula for will

become or


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From the figure above, , thus, the distance between cg and cp is

Eccentricity,

Total Hydrostatic Force on Curved Surfaces In the case of curved surface submerged in liquid at rest, it is more convenient to deal with thehorizontal and vertical components of the total force acting on the surface. Note: the discussion hereis also applicable to plane surfaces.

Horizontal Component The horizontal component of the total hydrostatic force on any surface is equal to the pressure onthe vertical projection of that surface.

Vertical Component The vertical component of the total hydrostatic force on any surface is equal to the weight of either real or imaginary liquid above it.

Total Hydrostatic Force

Direction of

Case 1: Liquid is above the curve surface The vertical component of the hydrostatic force is downward and equal to the volume of the realliquid above the submerged surface.


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Case 2: Liquid is below the curve surface The vertical component of the hydrostatic force is going upward and equal to the volume of theimaginary liquid above the surface.


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Analysis of Gravity Dam

Tags:

eccentricity

righting moment

overturning moment

foundation pressure

hydrostatic force

factor of safety

factor of safety against overturning

Factor of safety against sliding

friction

hydrostatic uplift

sliding

Dams are structures whose purpose is to raise the water level on the upstream side of river, stream,or other waterway. The rising water will cause hydrostatic force which will tend the dam to slidehorizontally and overturn about its downstream edge or toe. The raised water level on the upstreamedge or heel will also cause the water to seep under the dam. The pressure due to this seepage iscommonly called hydrostatic uplift and will reduce the stability of the dam against sliding and againstoverturning.

Gravity Dam Analysis The weight of gravity dam will cause a moment opposite to the overturning moment and the frictionon the base will prevent the dam from sliding. The dam may also be prevented from sliding bykeying its base into the bedrock.
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Step 1 Consider 1 unit length (1 m length) of dam perpendicular to the cross section.

Step 2 Determine all the forces acting:

1. Vertical forceso = Weight of damo = Weight of water in the upstream side (if any)

o = Hydrostatic uplifto Weight of permanent structures on the dam

2. Horizontal forceso = Horizontal component of total hydrostatic forceo Wind pressure, wave action, floating bodies, earthquake load, etc.


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If , is within the middle third and the foundation pressure is trapezoidal (triangular if isexactly ) acting from heel to toe.

For the sign of , use (+) at point where is nearest. From the diagram above, use (+)for and (-) for . A negative indicates compressive stress and a positive indicates tensilestress. A positive will occur when . In foundation design, soil is not allowed to carry tensilestress, thus, any will be neglected in the analysis.

If , is outside the middle third and the foundation pressure is triangular.


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Buoyancy

Tags:

Archimedes Principle

area submerged

buoyant force

draft

floating body

volume displaced

Archimedes Principle
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Archimedes (287-212 B.C.)

Any body immersed in a fluid is subjected to an upward force called buoyant force equal to theweight of the displaced fluid.

Where= buoyant force

= unit weight of fluid= volume of fluid displaced by the body

Buoyant force acting on a body submerged in fluid is merely the resultant of two vertical hydrostaticforces. Consider the cylindrical body shown below to have some length perpendicular to thedrawing. The horizontal components of hydrostatic force acting on the body are in equilibriumbecause the vertical projection of the body in opposite sides is the same.


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The upward force is the total force exerted by the fluid on the under surface of the body; thedownward force is the total force exerted by the fluid on the upper surface of the body. Sinceliquid pressure increases by depth, is greater than . The difference istherefore upward, and this difference is the buoyant force.

For homogeneous body of volume "floating" in a homogeneous liquid at rest, the volumedisplaced is

For a floating body of height and constant cross-sectional areaparallel to the liquid surface, the submerged length is given by

For a floating body whose cross-sectional area is perpendicular tothe liquid surface, the area submerged is given by

Stability of Floating Bodies

Tags:

righting moment

overturning moment

buoyant force
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floating body

volume displaced

center of buoyancy

metacenter

metacentric height

wedge of emersion

wedge of immersion

Any floating body is subjected by two opposing vertical forces. One is the body's weight W which isdownward, and the other is the buoyant force BF which is upward. The weight is acting at the center of gravity G and the buoyant force is acting at the center of buoyancy BO. W and BF are alwaysequal and if these forces are collinear, the body will be in upright position as shown below.

The body may tilt from many causes like wind or wave action causing the center of buoyancy to shiftto a new position as shown below.
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Tags:

tangential stress

circumferential stress

longitudinal stress

cylindrical tank

spherical tank

longitudinal section

spacing of hoops

transverse section

wood stave vessel

The circumferential stress, also known as tangential stress, in a tank or pipe can be determined byapplying the concept of fluid pressure against curved surfaces. The wall of a tank or pipe carrying

fluid under pressure is subjected to tensile forces across its longitudinal and transverse sections.

Tangential Stress, t (Circumferential Stress) Consider the tank shown being subjected to an internal pressure p. The length of the tank is Lperpendicular to the drawing and the wall thickness is t. Isolating the right half of the tank:
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Longitudinal Stress, l At the end of the tank, the total stress P T = L A end should equal the total fluid force F at that end.Since the wall thickness t is so small compared to internal diameter D, the area A end of the wall isclose to Dt.

Observe that the tangential stress is twice that of the longitudinal stress.

Spherical Shell If a spherical tank of diameter D and thickness t contains gas under a pressure of p, the stress at thewall can be expressed as:

Spacing of Hoops of Wood Stave Vessels It is assumed that the wood will not resist tension, only the hoops will resist all the tensile stresscaused by the internal pressure p.


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wheres = spacing of hoopst = allowable tensile stress of the hoop

Ah = cross-sectional area of the hoopp = internal pressure in the vesselD = internal diameter of the vessel

Relative Equilibrium of Liquids

Relative equilibrium of liquid is a condition where the whole mass of liquid including the vessel in

which the liquid is contained, is moving at uniform accelerated motion with respect to the earth, butevery particle of liquid have no relative motion between each other. There are two cases of relativeequilibrium that will be discussed in this section: linear translation and rotation. Note that if a mass of liquid is moving with constant speed, the conditions are the same as static liquid in the previoussections.


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Formulas For details of the following formulas see the translation and rotation pages.

Horizontal Motion

Inclined Motion

Vertical Motion

Rotation

and

Rectilinear Translation | Moving Vessel

Tags:

force polygon

constant acceleration

horizontal motion

inclined motion

vertical motion

fluid mass

Horizontal Motion If a mass of fluid moves horizontally along a straight line at constant acceleration a, the liquid surfaceassume an angle with the horizontal, see figure below.
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For any value of a, the angle can be found by considering a fluid particle of mass m on the surface. The forces acting on the particle are the weight W = mg , inertiaforce or reverse effective force REF = ma , and the normal force N which is the

perpendicular reaction at the surface. These three forces are in equilibrium with their force polygon shown to the right.

From the force triangle

Inclined Motion Consider a mass of fluid being accelerated up an incline from horizontal. The horizontal andvertical components of inertia force REF would be respectively, x = ma h and y = ma v.


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From the force triangle above

but a cos = a h and a sin = a v, hence

Use (+) sign for upward motion and (-) sign for downward motion.

Vertical Motion The figure shown to the right is a mass of liquid moving vertically upward witha constant acceleration a. The forces acting to a liquid column of depth h fromthe surface are weight of the liquid W = V , the inertia force REF = ma , and thepressure F = pA at the bottom of the column.

Use (+) sign for upward motion and (-) sign for downward motion. Also note that a is positive for acceleration and negative for deceleration.


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Rotation | Rotating Vessel

Tags:

centrifugal force

angular velocity

paraboloid

squared property of parabola

centripetal acceleration

slope of paraboloid

volume of paraboloid

When at rest, the surface of mass of liquid is horizontal at PQas shown in the figure. When this mass of liquid is rotatedabout a vertical axis at constant angular velocity radian per second, it will assume the surface ABC which is parabolic.Every particle is subjected to centripetal force or centrifugalforce CF = m 2x which produces centripetal accelerationtowards the center of rotation. Other forces that acts aregravity force W = mg and normal force N .

Where tan is the slope at the surface of paraboloid at any distance x from the axis of rotation.

From Calculus, y = slope, thus
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For cylindrical vessel of radius r revolved about its vertical axis, the height h of paraboloid is

Other Formulas By squared-property of parabola, the relationship of y, x, h and r is defined by

Volume of paraboloid of revolution

Important conversion factor

Documents

Hydrostatic Pressure on Surfaces